"Why would ice melting cause the sea level to rise? Ice takes up more room than water. When the ice in my tea melts the glass doesn't overflow. :^)"
Because the ice is above sea level.
If you put enough ice cubes in a glass so that the top ones are above the liquid level then the liquid level will rise as the ice melts.
For floating ice, that's not true. If the ice is floating on the water, as in the case of the Arctic, water-level should not over-flow.
Antarctica ice is a different scenario.
---If you put enough ice cubes in a glass so that the top ones are above the liquid level then the liquid level will rise as the ice melts.---
It seems to me that the ice displaces it's weight in water, so that the part sticking up represents the added volume it assumes in a frozen state. When it melts it should occupy the same volume as it displaced in water when it was frozen. No rise in levels.
If the ice is displacing water it will always be "above" the liquid level, at least part of it. If it is floating in the ocean it will displace water and when it melts it will not cause the level of the ocean to rise, in fact the oceans level would actually go down somewhat if all floating ice were to melt. It is not possible to fill the ocean up with ice, ice floats and it would be rather difficult to stack ice in the ocean the same way you can stack ice cubes into a glass before putting liquid into it.
To sum it up, ocean ice, that is ice floating in the ocean, will not add to sea volume when it melts.